In a previous __ post __ we looked at alternative investing through the prism of asymmetric payoffs. Hedge funds, a classic alternative investment vehicle, tend to have a concave payoff structure that is akin to a covered call strategy – the upside is capped while the downside is shared with the market. While the overall beta of hedge funds is low (~ 0.34), the beta value increases during significant negative markets and falls sharply during significant up markets. Similarly, hedge fund returns have a correlation of about 0.7 with the S&P 500 during significant down markets, but this drops to about 0.2 when markets rise. This is important because the volatility and diversification benefits that alternative strategies like hedge funds offer come from altering the payoff structure when markets go up, as opposed to taking different risks.

We closed by asking the question: why not utilize a strategy that has convex payoffs? The simplest way to construct one of these would be to buy protective puts in a long equity strategy to protect against large down markets. The problem is that tail-risk protection strategies carry a premium cost that can severely detract from long-term returns. In this post, we will look into what these costs are.

A recent paper by Israelov and Nielsen at AQR Capital Management ‘Still Not Cheap: Portfolio Protection in Calm Markets‘ ( __ link __ ) showed that seemingly cheap put options only give the illusion of value. The authors make an important point:

“Buying an option is not a bet that realized volatility will increase; it is a bet that realized volatility will increase above the option’s implied volatility. Buying an option is expected to lose money even when volatility is low and rising if the spread between realized and implied volatility is sufficiently high.”

### Portfolio Hedging using Put Options

We consider a strategy that is long the highly liquid __ SPY ETF __ (which replicates the S&P 500 Index) and is also long a 5% or 10% out-of-the-money (OTM) front-month SPY put option to protect the portfolio against a large downside shock. We use SPY options since these are more liquid and typically have a lower bid-ask spread than S&P 500 index options. The options data is only available from January 2005 and so the strategy is tested between the period February 2005 and September 2015.

The put options are rolled over on the last trading day of each month, and the portfolio is rebalanced at close prices. For example, at the end of December, the strategy would go long an out-of-the-money put option that expires in February. This put would be sold at the end of January while simultaneously going long a put option that expires in March. To make the returns more realistic, we buy the options at their ask price and sell at the bid. The portfolio is sized to unit leverage and the amount of put options in the portfolio hedges 95%/90% of the SPY position at the time of purchase. As an example: if the price of 1 share of SPY ETF is $198 and a 5% OTM put option on SPY costs $2, the percentage of portfolio allocated to the option is 2/(198+2) = 1%. The remaining 99% of the portfolio is assumed to be long the underlying SPY ETF.

As one would expect, a strategy that is long equity and also long an out-of-the-money put does result in an asymmetric, convex payoff. The two graphs that follow show scatter plots of monthly returns for the Long SPY + Long OTM Put (5% or 10%) strategy versus S&P 500 returns.

The best-fit curve in both cases show a convex profile with positive coefficients for the quadratic term. The goodness of fit (R-squared) term is higher than that obtained when we try a linear fit to the data. Note that the asymmetry is more pronounced when using 5% OTM puts for protection, which has a larger quadratic coefficient. The betas across the entire period for the SPY + 5% OTM Put and the SPY + 10% OTM Put strategies are 0.70 and 0.85, respectively. As you go further out-of-the-money, it costs less to protect your portfolio against a large loss and the strategy starts to look more similar to a long SPY only strategy.

The convexity of the payoff becomes more clear when looking at realized betas for the two strategies in three different market environments – when the S&P 500 falls more than 1.5% in a month, rises more than 1.5% in a month and is flat (between -1.5% and 1.5% monthly return). We see in the graph below that both strategies have more exposure to the market on the upside than the downside. For the SPY + 5% OTM Put strategy the beta rises from 0.46 in down markets to 0.62 in up markets, while for the SPY + 10% OTM Put strategy the beta rises from 0.68 in down markets to 0.75 in up markets.

Interestingly, the realized beta for both strategies is highest during flat markets as opposed to the strongest up markets. This is due to the fact that a lot of the large positive months for the S&P 500 occur right after the large negative months. However, that is when implied volatility is highest – essentially, the cost of protection rises sharply when the market falls and this drags down returns during some of the strongest months.

That brings us right into the question of costs. As we saw above, protecting a long equity portfolio with puts results in a seemingly attractive payoff structure. But this is obviously not free. The following graph shows the cumulative performance for both strategies between February 2005 and September 2015. The strategy that is only long SPY (buy and hold) is also included for comparison.

The overall return for a buy and hold SPY ETF strategy is 101.5% across the entire ten and half year period. However, if you bought protection each month to insure the portfolio against losses larger than 5%, you would have barely made any money: the overall return for the SPY + 5% OTM Put strategy is only 12%. The SPY + 10% OTM Put strategy fares a little better since the these puts cost less but the overall return of 38.5% is still significantly lower than buy and hold. Also note that these returns are before you factor in the transaction costs of rolling over put options each month.

The overwhelming cost of a put protection strategy can be seen in the following table detailing summary statistics and annual returns for the three strategies.

The period we consider here has not been a stellar one for equities, with the annualized return for buying and holding the SPY ETF only 6.8%. On the other hand, the annual returns for the SPY + 5% OTM Put and SPY + 10% OTM Put strategies are much lower, 1.1% and 3.1%, respectively. Buying protective puts would have extracted significant costs from the portfolio. Also, both protection strategies do not help in reducing the maximum drawdown (monthly peak-to-trough). The SPY + 5% OTM Put strategy has a drawdown of almost 45%, which is not very different from the 51% drawdown that the buy and hold strategy experienced.

In every year but for 2008, the protection strategies had lower returns than buy and hold. The SPY + 5% OTM Put strategy under-performs by an average of 6.3% annually while the SPY + 10% OTM Put strategy under-performs by an average of almost 4%. The under-performance is notable even in 2005 and 2006, when markets were relatively calm and volatility was low. The VIX index, which is the CBOE’s market expectation of near-term volatility as conveyed by S&P 500 Index option prices, averaged less than 13% in these two years.

Since the SPY tracks the S&P 500 Index closely, the VIX is essentially a measure of how ‘expensive’ or ‘cheap’ the SPY options we use in our protection strategies are. To understand how the two strategies perform in different regimes of volatility, we sort our strategy returns over the entire ten and half year historical period by quartiles of VIX (as of close on the rollover dates). The following graph shows the average annualized return of both strategies over buy and hold in each month.

The two strategies under-perform buy and hold in all VIX quartiles. The under-performance is greatest in the highest quartile (VIX averages 32%), but this is expected since options are most likely to be over-priced when volatility is relatively high. However, the under-performance is quite significant even in the lowest quartile, when VIX averages 12%: the SPY + 5% OTM Put strategy averages a return that is lower than buy and hold by 2.5% while the SPY + 10% OTM Put strategy under-performs by an average 1.2%.

The results presented above indicate that protecting a portfolio using puts to generate a convex payoff comes at significant cost. Protective puts have been costly even under benign market environments, i.e. when volatility is relatively low, and have not been ‘cheap’ in any sense. Market environments, and volatility regimes in particular, tend to persist. So while one may believe that a protective put can be bought for cheap, these puts will have to be bought several times over before it actually pays off. Even in that case, as we saw with our protection strategy returns in 2008 and overall drawdown levels, the price of protection jumps sharply when markets fall.

As __ Israelov and Nielsen __ write in their paper:

### “Proponents of buying put options in calm environments describe an opportunity to obtain protection at reduced prices during the calm before the storm. If history is any guide, the more likely outcome is that we are in the midst of the calm before the calm.”

Their study, along with ours, indicate that the historical evidence appears to favor selling over buying options, even in calm periods – this strategy does result in concave payoffs but the seller is compensated since they are taking downside market risk, an important point we discussed in a __ prior post __ about alternative investing. As the authors argue, purchasing put options, and their associated costs, may be rationalized if you believe that black swan events (think October 1987 when the market fell 20% in a day) are under-represented in the historical data and that put options are actually less expensive that they appear. However in that case the logical conclusion would be to allocate less of the portfolio to equities and more to uncorrelated or inversely correlated assets like fixed income.